rto-en-avt-162 17 - 1 · 13. supplementary notes see also ada562449. rto-en-avt-162,...

Qua nt i fi ca t i o n o f Uncer t a i nty i n Fl ow Si mul a t i o ns

Us i ng Pr o ba bi l i s t i c Met ho ds

Gianluca IaccarinoMechanical Engineering

Institute for Computational Mathematical Engineering

Stanford University

ABabcdfghiejklNon-equilibrium gas dynamics

From physical models to hypersonic flights

VKI Lecture Series - Sept. 8-12, 2008

RTO-EN-AVT-162 17 - 1

Report Documentation Page Form ApprovedOMB No. 0704-0188

Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering andmaintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information,including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, ArlingtonVA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if itdoes not display a currently valid OMB control number.

1. REPORT DATE SEP 2009

2. REPORT TYPE N/A

3. DATES COVERED -

4. TITLE AND SUBTITLE Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S) 5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Mechanical Engineering Institute for Computational MathematicalEngineering Stanford University

8. PERFORMING ORGANIZATIONREPORT NUMBER

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)

11. SPONSOR/MONITOR’S REPORT NUMBER(S)

12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release, distribution unlimited

13. SUPPLEMENTARY NOTES See also ADA562449. RTO-EN-AVT-162, Non-Equilibrium Gas Dynamics - From Physical Models toHypersonic Flights (Dynamique des gaz non- equilibres - Des modeles physiques jusqu’au volhypersonique)., The original document contains color images.

14. ABSTRACT

15. SUBJECT TERMS

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT

SAR

18. NUMBEROF PAGES

30

19a. NAME OFRESPONSIBLE PERSON

a. REPORT unclassified

b. ABSTRACT unclassified

c. THIS PAGE unclassified

Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

Contents

1 Introduction and motivation 3

2 Definitions and basic concepts 4

2.1 Errors vs. uncertainties . . . . . . . . . . . . . . . . . . . . . 42.2 Aleatory uncertainty . . . . . . . . . . . . . . . . . . . . . . . 42.3 Epistemic uncertainty . . . . . . . . . . . . . . . . . . . . . . 52.4 Sensitivity vs. uncertainty analysis . . . . . . . . . . . . . . . 5

3 Predictions under uncertainty 6

3.1 Data assimilation . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Uncertainty propagation . . . . . . . . . . . . . . . . . . . . . 73.3 Certification . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Probabilistic uncertainty propagation 8

4.1 Sampling techniques . . . . . . . . . . . . . . . . . . . . . . . 84.1.1 Monte Carlo method . . . . . . . . . . . . . . . . . . . 94.1.2 Latin Hypecube approach . . . . . . . . . . . . . . . . 10

4.2 Quadrature methods . . . . . . . . . . . . . . . . . . . . . . . 104.2.1 Stochastic collocation . . . . . . . . . . . . . . . . . . 114.2.2 Extension to multiple dimensions . . . . . . . . . . . . 14

4.3 Spectral methods . . . . . . . . . . . . . . . . . . . . . . . . . 154.3.1 Stochastic Galerkin approach . . . . . . . . . . . . . . 154.3.2 Extension to multiple dimensions . . . . . . . . . . . . 17

5 Examples 18

5.1 Steady Burgers equation . . . . . . . . . . . . . . . . . . . . . 185.2 Heat transfer computations under uncertainty . . . . . . . . . 215.3 Shock dominated problems . . . . . . . . . . . . . . . . . . . 22

6 Conclusions and outlook 26

Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods

17 - 2 RTO-EN-AVT-162

1 Introduction and motivation

In the last three decades, computer simulation tools have achieved widespread use in the design and analysis of engineering devices. This hasshortened the overall product design cycle and it has also provided bet-ter understanding of the operating behavior of the systems of interest. Asa consequence numerical simulations have lead to a reduction of physicalprototyping and to lower costs.

In spite of this considerable success, it remains difficult to provide objec-tive confidence levels in quantitative information obtained from numericalpredictions. The complexity arises from the amount of uncertainties relatedto the inputs of any computation attempting to represent a physical system.As a result, especially in the area of reliability and safety, physical testingremains the dominant mechanism of certification of new devices. Rigor-ous quantification of the errors and uncertainties1 introduced in numericalsimulations is required to establish objectively their predictive capabilities.

Procedures to establish the quality of numerical simulations have beenorganized within the framework of Verification and Validation (V&V) ac-tivities [34]. Verification is a mathematical process that aims at answeringthe question: “are we solving the equations correctly?”. The objective isto quantify the errors associated to the algorithms employed to obtain thesolution of the governing equations. Validation, on the other hand, aims atanswering the question “are we solving the correct equations?”. The goalis to identify the appropriateness of the selected mathematical/physical for-mulation to represent the device to be analyzed. Validation always involvescomparisons of the numerical predictions to reality, whereas verification only

involves numerical analysis and tests.There is a growing recognition of the fact that validation cannot be car-

ried out without explicitly accounting for the uncertainties present in boththe measurements and the computations. Experimentalists are typically re-quired to report uncertainty bars to clearly identify the repeatability and theerrors associated to the measurements. Validation must be carried out ac-knowledging the nature of the experimental uncertainties and by providinga similar indication of the computational error bars. One of the objectiveof uncertainty quantification methods is to construct a framework to esti-mate the error bars associated to given predictions. Another objective is toevaluate the likelihood of a certain outcome; this obviously leads to betterunderstanding of risks and improves the decision making process.

1The difference between errors and uncertainties will be given later


RTO-EN-AVT-162 17 - 3

2 Definitions and basic concepts

The uncertainty quantification community has introduced precise definitionsto characterize various types of uncertainties.

2.1 Errors vs. uncertainties

The American Institute of Aeronautics and Astronautics (AIAA) ”Guidefor the Verification and Validation of CFD Simulations” [35] defines errors

as recognisable deficiencies of the models or the algorithms employed anduncertainties as a potential deficiency that is due to lack of knowledge.

This definition is not completely satisfactory because does not preciselydistinguish between the mathematics and the physics. It is more useful todefine errors as associated to the translation of a mathematical formulationinto a numerical algorithm (and a computational code).

Errors are typically also further classified in two categories: acknowl-edged errors are known to be present but their effect on the results is deemednegligible. Examples are round-off errors and limited convergence of certainiterative algorithms. On the other end, unacknowledged errors are not rec-ognizable2 but might be present; implementation mistakes (bugs) or usageerrors can only be characterized by comprehensive verification tests andprocedures.

Using the present definition of errors, the uncertainties are naturally as-sociated to the choice of the physical models and to the specification of theinput parameters required for performing the analysis. As an example, nu-merical simulations require the precise specification of boundary conditionsand typically only limited information are available from corresponding ex-periments and observations. Therefore variability, vagueness, ambiguity andconfusion are all factors that introduce uncertainties in the simulations. Amore precise characterization is based on the distinction in aleatory andepistemic uncertainties.

2.2 Aleatory uncertainty

Aleatory uncertainty3 is the physical variability present in the system beinganalysed or its environment. It is not strictly due to a lack of knowledge

2In principle, using the AIAA definition, unacknowledged errors could be considereduncertainties because they are associated to lack of knowledge

3Aleatory uncertainty is also referred to as variability, stochastic uncertainty or irre-ducible uncertainty.


17 - 4 RTO-EN-AVT-162

and cannot be reduced. The determination of material properties or operat-ing conditions of a physical system typically leads to aleatory uncertainties;additional experimental characterization might provide more conclusive ev-idence of the variability but cannot eliminate it completely. Aleatory uncer-tainty is normally characterized using probabilistic approaches.

2.3 Epistemic uncertainty

Epistemic uncertainty4. is what is indicated in the AIAA Guide (AIAA1998) as ”uncertainty”5, i.e. a potential deficiency that is due to a lack ofknowledge. It can arise from assumptions introduced in the derivation ofthe mathematical model used or simplifications related to the correlation ordependence between physical processes. It is obviously possible to reduce theepistemic uncertainty by using, for example, a combination of calibration,inference from experimental observations and improvement of the physicalmodels. Epistemic uncertainty is not well characterized by probabilistic ap-proaches because it might be difficult to infer any statistical informationdue to the nominal lack of knowledge. A variety of approaches have beenintroduced to provide a more suitable framework for these analysis. Typi-cal examples of sources of epistemic uncertainties are turbulence modelingassumptions and surrogate chemical kinetics models.

2.4 Sensitivity vs. uncertainty analysis

Sensitivity analysis (SA) investigates the connection between inputs andoutputs of a (computational) model; more specifically, it allows to identifyhow the variability in an output quantity of interest is connected to aninput in the model and which input sources will dominate the response ofthe system. On the other hand, uncertainty analysis aims at identifying theoverall output uncertainty in a given system. The main difference is thatsensitivity analysis does not require input data uncertainty characterizationfrom a real device; it can be conducted purely based on the mathematicalform of the model. As a conclusion large output sensitivities (identifiedusing SA) do not necessarily translate in important uncertainties becausethe input uncertainty might be very small in a device of interest.

SA is often based on the concept of sensitivity derivatives, the gradient ofthe output of interest with respect to input variables. The overall sensitivity

4Epistemic uncertainty is also called reducible uncertainty or incertitude5Aleatory uncertainty is not mentioned in the AIAA Guidelines


RTO-EN-AVT-162 17 - 5

is then evaluated using a Taylor series expansion, which, to first order, wouldbe equivalent to a linear relationship between inputs and outputs.

3 Predictions under uncertainty

Computer simulations of an engineering device are performed following asequence of steps.

Initially the system of interest and desired performance measures aredefined. The geometrical characterization of the device, its operating condi-tions, the physical processes involved are identified and their relative impor-tance must be quantified. It is worthwhile to point out that the definition ofthe system response of interest is a fundamental aspect of this phase. Thenext step is the formulation of a mathematical representation of the system.The governing equations and the phenomenological models required to cap-ture the relevant physical processes need to be defined. In addition, the pre-cise geometrical definition of the device is introduced. This step introducessimplification with respect to the real system; for example small geometricalcomponents are eliminated, or artificial boundaries are introduced to reducethe scope of the analysis. With a well defined mathematical representationof the system, the next step if to formulate a discretized representation.Numerical methods are devised to convert the continuous form of the gov-erning equations into an algorithm that produces the solution. This steptypically requires, for example, the generation computational grid, which isa tessellation of the physical domain. Finally the analysis can be carriedout.

The introduction of uncertainty in numerical simulations does not alterthis process but introduces considerable difficulties in each phase. It is usefulto distinguish three steps: data assimilation, uncertainty propagation andcertification.

3.1 Data assimilation

Data assimilation consists of a study of the system of interest that aims atidentifying the properties, physical processes and other factors required tofully characterize it. The analysis is typically focused on the specific in-puts required by the mathematical framework that will be applied in thesimulations. As an example, the boundary conditions required in numeri-cal simulations should be inferred from observation of the device of inter-est or specific experiments. Given the limited degree of reproducibility ofexperimental measurements and the errors associated to the measurement


17 - 6 RTO-EN-AVT-162

techniques, these quantity are known with a certain degree of uncertainty(typically specified as an interval, x ± y%). Probabilistic approaches treatthese quantities, that overall characterize the aleatory uncertainty, as ran-

dom variables assuming values within specified intervals. In mathematicalterms this is equivalent to define random variables with a specified probabil-ity distribution functions (PDF) [13]. The obvious choice is to use randomvariables defined using analytical distributions (Gaussian, uniform, etc.). Itis difficult to justify this choice [5] solely from experimental evidence be-cause of the limited amount of data typically available6; in many situationthe only data available are obtained from expert opinions and can lead toambiguous or conflicting estimates. Alternative approaches have been de-vised to provide a more flexible framework to handle this situation, evidencetheory is one such approach [37, 38].

In the context of probabilistic approaches, the objective of data assimila-tion is to define PDFs of each of the input quantity used in the computationaltool.

3.2 Uncertainty propagation

Once probability distributions are available for all the input quantities inthe computational algorithm, the objective is to compute the PDFs of theoutput quantities of interest. This step is usually the most complex andcomputationally intensive for realistic engineering simulations. A variety ofmethods are available in the literature, from sampling based approaches (e.g.Monte Carlo) to more sophysticated stochastic spectral Galerkin approaches.In the next section these methods are described in detail.

3.3 Certification

Once the statistics of the quantity of interest have been computed, variousmetrics can be used to characterize the system output, depending on thespecific application. The most common use of such statistical informationis a reliability assessments, where the likelihood of a certain outcome is es-timated and compared to operating margins. In a validation context, thePDF (or more typically the cumulative distribution function) is compared toexperimental observation to extract a measure of the confidence in the nu-merical results. The characterization of these measures, so-called validation

6Note that the specification of an interval is not equivalent to a specification of auniform probability distribution. An interval formally does not contain probabilistic in-formation!


RTO-EN-AVT-162 17 - 7

metrics, is an active area of research [33].

4 Probabilistic uncertainty propagation

Within a probabilistic framework, the problem of uncertainty propagationconsists of the generation of PDFs of the outcomes given (known) distri-bution of all the input parameters. Several classes of methods have beendeveloped to solve this problems; in this section three popular approachesare described.

Consider the vector x = (x1, ...xD) containing the input quantities tothe computational model; assume that y = g(x) is the output of interest; gis possibly the result of a complex fluid dynamic simulation.

In probabilistic uncertainty quantification approaches the stochastic, in-put quantities x are represented as independent continuous random vari-ables xi(ωi)

7 mapping the sample space Ωi to real numbers, xi : Ωi → R.This assumption in practical terms increases the dimensionality of the prob-lem: the original deterministic outcome y = f(x1, ..., xi, ...xD) becomes astochastic quantity y = f(x1, ..., xi, ...xD : ω1, ..., ωi, ..., ωD). The objectiveis to compute the PDF of y, fy, in order to evaluate the likelihood of acertain outcome, or, in general, statistics of y. The expected value E[y] andthe variance V ar[y] are defined as:

E[y] =

∫

∞

∞

zfy(z)dz (1)

V ar[y] =

∫

∞

∞

(z − E[y])2 fy(z)dz = E[y2] − (E[y])2 . (2)

Note that y is a stochastic variable while the expected value and the varianceare deterministic quantities.

4.1 Sampling techniques

Sampling-based techniques are the simplest approaches to propagate un-certainty in numerical simulations: they involve repeated simulations (alsocalled realizations) with a proper selection of the input values. All the resultsare then collected to generate a statistical characterization of the outcome.In the following the Monte Carlo approach and the Latin Hypercube sam-pling strategy are described.

7The assumption of independence is mathematically stated as fx1,...,xD= fx1

· · · fxD,

where fx1,...,xDthe joint PDF is the product of the marginals


17 - 8 RTO-EN-AVT-162

ξ

f ξ

-10 -5 0 5 10

Figure 1: Equally probability intervals defined on the input parameter spacefor a Gaussian PDF. Each of the interval is sampled in the Latin hypercubemethod.

4.1.1 Monte Carlo method

The Monte Carlo method [31] is the oldest and most popular sampling ap-proach. It involves random sampling from the space of the random variablesxi according to the given PDFs. The outcome is typically organized as ahistogram and the statistics are readily computed from Eq. (2) by replac-ing the integrals with sums over the number of samples. The method hasthe advantage that it is simple, universally applicable and does not requireany modification to the available (deterministic) computational tools. It isimportant to note that while the method converges to the exact stochasticsolution as the number of samples goes to infinity, the convergence of themean error estimate is slow. Hence thousands or millions of data samplesmay be required to obtained accurate estimations. However, the conver-gence does not directly depend on the number of random variables in theproblem. In this form the Monte Carlo methods always give the correctanswer, but a prohibitively large number of realizations may be required toaccurately estimate responses that have a small probability of occurrence.On the other hand, the convergence of the low order statistics (expectedvalue and variance) require much smaller number of samples.


RTO-EN-AVT-162 17 - 9

Figure 2: Sampling based techniques with two uncertain inputs. (left) MonteCarlo; (center) Latin Hypercube; (right) Lattice based

4.1.2 Latin Hypecube approach

Several methods have been developed to accelerate the Monte Carlo ap-proach. One of the most successful is the Latin Hypercube sampling (LHS)[29] approach. The range of each input random variable xi is divided inM intervals with equal probability. In Fig. 1 this is illustrated for a singleinput Gaussian random variable. M random samples are drawn, one fromeach of the intervals and the convergence is faster than Monte Carlo becausethe occurrence of low probability samples is reduced. The construction isespecially effective in multiple dimensions; in Fig. 2 an example of samplesgenerated using Monte Carlo and Latin Hypercube for two uniform ran-dom variables is reported. The LHS samples projected on each of the axisprovide optimal coverage of the parameter space [30]. On the other hand,the Monte Carlo samples show clusters and holes that eventually lead tothe slower convergence. In Fig. 2 another strategy is illustrated: Lattice-based sampling. In this approach, the parameter space is discretized usingregularly spaced points (lattice); once the solution is evaluated at those lo-cations a random shift is applied to the lattice and another set of solutionsare computed[31]. The choice of the lattice is related to the distribution ofthe input variables similarly to the LHS approach and the objective is againto reduce the occurrence of clusters and holes in the samples.

4.2 Quadrature methods

One of the objective of UQ propagation methods is the computation ofthe statistics of an outcome of interest, such as its expectation and thevariance. As shown earlier, these require the evaluation of integrals (over


17 - 10 RTO-EN-AVT-162

the parameter space) and it is therefore natural to employ conventional

numerical integration techniques8.Let’s consider a problem with one uncertain parameter ξ; the objective

is to compute integrals of y(ξ). A class of quadrature rules are based on in-terpolating basis functions that are easy to integrate, typically polynomials.The integral is expressed as a weighted sum of the integrand y evaluatedin a finite number of locations on the ξ-axis (abscissas). The choice of thepolynomial basis defines the weights and the corresponding abscissas.

The simplest example is the midpoint rule in which a sequence of Mequally spaced values ξ(i) in the interval [a, b] (x1 = a and xM = b) isconsidered. The function y is evaluated at M − 1 abscissas as y((ξ(i) +ξ(i+1))/2) = yi+1/2. The integral is simply obtained as the sum of the M −1values yi+1/2 times the size of the interval (b − a). In this case the basisfunctions are piecewise constant and the weights are all equal to (b− a)/M .Quadratures based on equally spaced abscissas include the commonly usedtrapezoidal and Simpson rules and are, in general, referred to as Newton-Cotes formulas.

4.2.1 Stochastic collocation

Stochastic collocation refers to quadrature methods used to compute in-tegrands of random variables, thus over a stochastic domain. AlthoughNewton-Cotes formulas are applicable in this context, it is usually prefer-able to consider more general approaches, in which the abscissas are notequally spaced. Gaussian quadratures are popular in the field of uncertaintyanalysis because of their high accuracy9; the general format is:

∫ b

ay(ξ)f(ξ)dξ =

M∑

k=0

w(ξk)y(ξk) + RM (y) (3)

where the abscissas ξk are associated with zeros of orthogonal polynomialsand wk are weights and f(ξ) is a weighting function; in the present context,f represents a PDF. RM is the remainder term which determines the orderof accuracy of the integration. In particular, the formula (3) is said to havepolynomial degree of exactness p if RM (y) = 0 when y(ξ) is a polynomialof degree ≤ p. Newton-Cotes formulae have p = M − 1 whereas Gaussian

8Numerical integration and quadrature are typically synonym for one dimensional func-tions. In higher dimensions the term cubature is often used instead of quadrature

9High order polynomial interpolating functions constructed on equally spaced gridsalso suffer from the Runge’s phenomenon.


RTO-EN-AVT-162 17 - 11

Figure 3: (Legendre (left) and Hermite (right) polynomials.

quadratures achieve higher p with the same number of function evaluations(M + 1), specifically p = 2M + 1. This is the main advantage of Guassianquadrature and is accomplished by virtue of the orthogonality condition ofthe interpolating basis functions[24] [4], which is in general stated as:

〈pm, pn〉 =

∫ b

apm(ξ)pn(ξ)f(ξ)dξ = δnm (4)

where δnm is the Kroneker symbol.The most commonly used form of Guassian quadrature is the Gauss-

Legendre integration formula which is based on Legendre polynomials ℓk(ξ).In Fig. 3 the first few polynomials are reported. These are constructedusing the three term recurrence relation (with ℓ0(ξ) = 1 and ℓ1(ξ) = ξ):

(n + 1)ℓn+1(ξ) = (2n + 1)ξℓn(ξ) − nℓn−1(ξ)

The weight function in Eq. (3) is simply f(ξ) = 1; the integrationweights are computed as

wk =

∫ b

aℓk(ξ)dξ

and the abscissas ξk for an M -point quadrature rule are the zeros of theℓM+1 polynomial. As an example the computation of expectation of anoutput in the presence of an uncertain input defined in a interval [a, b] (asa uniform random variable) can be computed as:

E(y) =

∫ b

ay(ξ)fy(ξ)dξ =

M∑

k=0

y(ξk)wk (5)


17 - 12 RTO-EN-AVT-162

For uncertain parameters defined on unbounded domains such as ran-dom variables described by Gaussian probability distributions, the Gauss-Hermite quadrature is used [19]. In this case the three term recurrencerelation is (h0(ξ) = 1 and h1(ξ) = ξ):

hn+1(ξ) = 2ξhn(ξ) − 2nhn−1(ξ)

and the weight function is f(ξ) = e−ξ2

. In this case the integration weightswk are expressed as:

wk =

∫

∞

−∞

hk(ξ)e−ξ2

dξ

and the expectation can be computed as before just evaluating the outcomeat M + 1 abscissas that correspond to the zeros of the Hermite polynomialhM+1.

There is an obvious connection between the choice of the polynomialand the probability distribution of the input variables: the orthogonalitycondition (Eq. 4). In the present context, in order to achieve the polyno-mial exactness of Gaussian quadratures for computing the expectation, thepolynomials have to be orthogonal with respect to the weight function thatcorresponds to the PDF of the input variable.

The two Gauss quadrature rules described above require the computationof a new set of abscissas (and corresponding function evaluations) for eachdesired order M ; alternative approaches have the property that the abscissasare nested, e.g. different accuracy formulas share abscissas. A popularnested formulation rule is the Clenshaw-Curtis quadrature[17].

In practical terms, collocation methods for uncertainty propagation re-quire the evaluation of zeros and weights for a family of orthogonal basisfunctions; these can be computed and stored in advance. A set of M + 1independent computations are performed and the results are combined toobtain the statistics of the output of interest. Collocation can thereforebe interpreted as a sampling technique; it retains the main advantage ofthe Monte Carlo method because it does not require modifications to theexisting computational tool.

The evaluation of the PDF of the output quantities is somewhat morecomplicated for stochastic collocation methods than the computations ofthe output statistics. The first step is the construction of the polynomialinterpolant of the solution y(ξ) in the parameter space which involves theuse of the M +1 solutions y(ξ(i)). At this point, the interpolant is used as a


RTO-EN-AVT-162 17 - 13

Figure 4: Tensor (left) and sparse (right) grid quadrature rules for two-dimensional functions.

replacement for the original function10 and Monte Carlo sampling is used.

4.2.2 Extension to multiple dimensions

The natural extension of interpolation techniques to multiple dimensionsis a tensor product of one-dimensional interpolants. Let G(ξ1, . . . , ξD) bethe outcome of interest, expressed as a function of D random variables.The set of points for the multi-dimensional interpolant is the tensor prod-uct of the abscissas for the one-dimensional interpolants; in the case thatthe polynomial order is the same in every direction, the total number ofabscissas is (M + 1)D. Although the tensor product extension of the one-dimensional quadrature rules is simple and accurate, it clearly suffers fromthe exponential increase in the number of abscissas - and consequently re-quired function evaluations - as the number of dimensions increases [20]. Tocombat this, some have proposed using an alternative extension based onthe sparse grid construction, attributed to Smolyak [15]. In this case themulti-dimensional product is constructed as a linear combination of tensorproduct interpolants, each with relatively few points in its respective pointset. This greatly reduces the total number of abscissas - particularly asthe dimension D increases - while retaining the accuracy and convergence

10The approximated function is typically called a surrogate response surface


17 - 14 RTO-EN-AVT-162

properties of one-dimensional interpolants.Fig. 4 illustrates the extension of Gauss-Legendre quadrature in two-

dimensions using tensor products and sparse grid rules. Different set ofabscissas (M is varied from 1 to 11) are reported to illustrate how, forsparse grids, the number of points and corresponding function evaluationsremains relatively low even for high order polynomial interpolants.

Sparse grid constructions reduce the curse of dimensionality and con-verge very rapidly if the function of interest is smooth. Precise convergenceresults and comparisons to Monte Carlo methods are reported in [9]. Exten-sions of this approach to construct anisotropic grids are currently an activearea of research; the objective in this case is to increase the order of theinterpolating polynomials only in selected directions to further reduce thecomputational cost.

Other integration methods in high dimensions are also available in theliterature, for example Stroud formulae [10] although have not yet beenexplored in the area of uncertainty quantification.

4.3 Spectral methods

The third class of methods for uncertainty propagation is based on spectralmethods in which the solution is expressed using a suitable series expansion.These approaches are intrusive in the sense that the mathematical formula-tion requires modification of the existing deterministic codes - in contrast tothe sampling approaches and the stochastic collocation methods describedbefore. For this reason, spectral methods have to be described with referenceto a given mathematical problem; in the following the Burgers equation isused.

4.3.1 Stochastic Galerkin approach

In this approach the unknown quantities are expresses as an infinite series oforthogonal polynomials defined on the space of he random input variable11.This representation has its roots in the work of Wiener [23] who expressed aGaussian process as an infinite series of Hermite polynomials. Ghanem andSpanos [22] truncated Wiener’s representation and used the resulting finiteseries as a key ingredient in a stochastic finite element method.

As an example, consider the 1D Burgers equations

ut + uux = 0

11Stochastic Galerkin methods based on polynomial expansions are often referred to asPolynomial Choas approaches [22]


RTO-EN-AVT-162 17 - 15

and assume that the initial condition is uncertain and characterized by oneparameter, ξ a Guassian random variable.

u(x, t = 0) = ξ ∗ sin(x)

In stochastic Galerkin approaches the (unknown) solution u = u(x, t, ξ)is expressed as a spectral expansion as

u(x, t, ξ) =

∞∑

i=0

ui(x, t)Ψi(ξ) (6)

where the coefficients ui(x, t) are independent of the random variable and,therefore, deterministic (they are still function of the physical dimensions).The Ψi(ξ) are Hermite polynomials and form a complete orthogonal set ofbasis functions.

The expansion (6) is truncated to a certain order M and the approxi-mated expression for u is inserted in the governing equation:

M∑

i=0

∂ui

∂tΨi(ξ) +

M∑

j=0

ujΨj(ξ)

(

M∑

i=0

∂ui

∂xΨi(ξ)

)

= 0

Multiplying by Ψk(ξ) and integrating over the probability space - Galerkinprojection - we obtain a system of M coupled & deterministic equations forthe coefficients ui:

M∑

i=0

∂ui

∂t〈ΨiΨk〉 +

M∑

i=0

M∑

j=0

ui∂uj

∂x〈ΨiΨjΨk〉 = 0 for k = 0, 1, ...,M. (7)

Note that in Eq. (7) the double and triple products of the Hermitepolynomials are simply numbers [24] (cfr. Eq. 4):

〈ΨiΨj〉 = δiji!

〈ΨiΨjΨk〉 = 0 if i + j + k is odd or max(i, j, k) > s

〈ΨiΨjΨk〉 =i!j!k!

(s − i)!(s − j)!(s − k)!otherwise

and s = (i + j + k)/2.The statistics of the solutions can be readily computed from the expansions

coefficients. In particular the expectation is


17 - 16 RTO-EN-AVT-162

E[u] = E

[

M∑

k=0

ukΨk

]

= u0E[Ψ0] +

M∑

k=1

ukE[uk] = u0

for the orthogonality of the polynomial basis. The variance is computed as:

V ar[u] = E[

(u − E[u])2]

= E

((

M∑

k=0

ukΨk

)

− u0

)2

=

E

(

M∑

k=1

ukΨk

)2

=

M∑

k=1

u2kE[

Ψ2k

]

=

M∑

k=1

u2k(k!)2

As an example, the polynomial chaos formulations for the Burgers equationscan be written explicitly. For M = 2 the expansion (6) is u(x, t, ξ) = u0 + ξu1 andthe governing system of equations is:

∂u0

∂t+ u0

∂u0

∂x+ u1

∂u1

∂x= 0

∂u1

∂t+ u1

∂u0

∂x+ u0

∂u1

∂x= 0

For M = 3 the expansion (6) is u(x, t, ξ) = u0 + ξu1 + 2(ξ2 − 1)u2 and thegoverning system of equations is:

∂u0

∂t+ u0

∂u0

∂x+ u1

∂u1

∂x+ 2u2

∂u2

∂x= 0

∂u1

∂t+ u1

∂u0

∂x+ (u0 + 2u2)

∂u1

∂x+ 2u1

∂u2

∂x= 0

∂u2

∂t+ u2

∂u0

∂x+ u1

∂u1

∂x+ (u0 + 4u2)

∂u2

∂x= 0

It is worth to note that the choice of the polynomial basis is again connectedto the distribution of the input random variables. As for stochastic collocationapproaches Hermite and Legendre polynomials are used for Gaussian and uniformrandom variables respectively. A general formulation based on the Askey polyno-mials is presented in [21].

While the stochastic Galerkin technique has been shown to produce highly ac-curate statistics, it does requires extensive modifications to existing computationalmethods to solve for the coefficients of the expansion.

4.3.2 Extension to multiple dimensions

The presence of multiple uncertain parameters increases considerably the computa-tional complexity of the stochastic Galerkin approach although it does not changethe overall formulation.


RTO-EN-AVT-162 17 - 17

It is formally possible to formulate the basis polynomials in terms of a fulltensor product of one-dimensional polynomials similarly to what considered for thestochastic collocation approach. Ghanem and Spanos [22] introduced a differentexpansion. For example for a problem with two uncertain parameters expressed interms of Gaussian random variables the first few bi-variate Hermite polynomialsare:

Φ0(ξ1, ξ2) = 1 (8)

Φ1(ξ1, ξ2) = ξ1

Φ2(ξ1, ξ2) = ξ2

Φ3(ξ1, ξ2) = ξ21 − 1

Φ4(ξ1, ξ2) = ξ1ξ2

Φ5(ξ1, ξ2) = ξ22 − 1

In multiple dimensions the number of coefficients M is determined by the num-ber of dimensions D and the highest polynomial order P :

M =(D + P )!

D!P !(9)

As in stochastic collocation approach, the stochastic Galerkin method suffersfrom the curse of dimensionality. Smolyak-type constructions have not been at-tempted for the polynomial basis used in the spectral expansions but offer a possibleremedy.

5 Examples

The examples reported in the following offer indications of the relative computa-tional cost of the uncertainty propagation approaches illustrated before. In thesetest cases the uncertainty is assumed as part of the problem specification.

5.1 Steady Burgers equation

The first test problem is based on the 1D viscous Burgers equations [36]:

1

2(1 − u)

∂u

∂x= µ

∂2u

∂u2

where the convective flux is modified so that an exact solution can be obtained(manufactured solution [34]):

u(x) =1

2

[

1 + tanh

(

x

4µ

)]


17 - 18 RTO-EN-AVT-162

x

E[u

]

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

MC 10MC 100MC 1000MC 10000

x

u

0.9 1 1.1 1.2 1.30.9

0.92

0.94

x

E[u

]

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

MC 10MC 100MC 1000MC 10000

x

u

0.9 1 1.1 1.2 1.30.9

0.92

0.94

Figure 5: Expectation of the solution for the Burgers steady state problemwith uncertain viscosity. Monte Carlo sampling with different number ofrealization. A numerical solution computed using 32 cells (left) is comparedto the exact solution (right).

In this case it is assumed that the viscosity is uncertain, for example as aconsequence to an unknown fluid temperature. A Gaussian random variable isconsidered with mean and variance given by E[µ] = 0.25 and var[µ] = 0.002512

Calculations have been performed using a finite volume discretization on thedomain [-3:3]; boundary conditions are derived from the exact solution. MonteCarlo sampling is initially used as the uncertainty propagation technique. Theexpectation of the velocity is reported in Fig. 5.1 and has been computed usinga different number of samples ranging from 10 to 10,000. The results shows onlysmall dependency on the number of realization; in the same figure the statisticalconvergence is also evaluated using the exact solution of the deterministic problem.In Fig. 5.1 the solution variance is reported. In this case it is clear that 10 samplesare not sufficient and converged results are only obtained for 1,000 and 10,000samples. Note that the variance is zero at x = 0 to satisfy the symmetry constraint.

As a second step the stochastic Galerkin method has been implemented andtested. In this case the expansion is based on Hermite polynomials. Computationshave been performed using the same discretization used in the deterministic code[36]. Fig. 5.1 shows results obtained using different order of expansions, from 1 to 3;the comparisons to Monte Carlo show that for this simple problem - characterized bysmall variability - even the low order expansion provide a reasonable representation

12It should be noted that a Gaussian distribution for the viscosity implies that there isa non-zero probability of having very low - or even negative - values of the viscosity. Inthis case the high mean value and the small variance reduce the occurrence of negativeviscosity to virtually zero.


RTO-EN-AVT-162 17 - 19

x

Var

[u]

-3 -2 -1 0 1 2 30

0.001

0.002

0.003

MC 10MC 100MC 1000MC 10000

x

Var

[u]

-3 -2 -1 0 1 2 30

0.001

0.002

0.003

MC 10MC 100MC 1000MC 10000

Figure 6: Variance of the solution for the Burgers steady state problemwith uncertain viscosity. Monte Carlo sampling with different number ofrealization. A numerical solution computed using 32 cells (left) is comparedto the exct solution (right).

x

E[u

]

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

MC 1000SG M=1SG M=2SG M=3

x

var[

u]

-3 -2 -1 0 1 2 30

0.001

0.002

0.003

MC 1000SG M=1SG M=2SG M=3

Figure 7: Expectation (left) and variance (right) of the solution for the Burg-ers steady state problem with uncertain viscosity. The Stochastic Galerkinmethod is used and the results are compared to Monte Carlo sampling.


17 - 20 RTO-EN-AVT-162

Figure 8: Computational grid for the flow around an array of cylinders withuncertain surface heat flux

of the variance of the solution.

5.2 Heat transfer computations under uncertainty

In this problem the goal is the computation of the wall temperature for an array ofcylinders in crossflow under uncertain free stream conditions and surface heat flux.The mathematical framework is based on the two-dimensional Reynolds-averagedNavier-Stokes equations and the computational domain is a channel with a specifiedinflow, periodic boundary conditions on the top and bottom boundaries, and anoutflow. An unstructured grid is used, see Fig. 5.2.

We assume that the sources of uncertainty are the specification of the velocityboundary condition on the incoming flow and the definition of the thermal conditionon the surface of the cylinder. The inlet velocity profile is constructed as a linearcombination of two cosine functions:

Uinlet(y) = 1 + 0.25(ξ1 cos(2πy) + ξ2 cos(10πy)). (10)

where ξ1 and ξ2 are two uniform random variables in the interval [-1:1]; this inflowspecification ensures that the amplitude of the random fluctuations did not causethe velocity to be negative. The wave numbers 2 and 10 were chosen to give onelow and one high frequency fluctuation while maintaining symmetry in the problem.The heat flux is specified as an exponential function of ξ3 over the cylinder, namely

∂T

∂n cylinder(x) = e−(0.1+0.05ξ3)(x−0.5) (11)

where n is the normal to the cylinder surface. The heat flux is slightly greaterat the left side of the cylinder at the stagnation point; the value of ξ3 determinesprecisely how much greater. Again ξ3 is a uniform random variable in [-1:1].

We employed Monte Carlo sampling and stochastic collocation to solve thisproblem. In particular, we used both tensor product and sparse grid formalism to


RTO-EN-AVT-162 17 - 21

l # tensor grid points # sparse grid points

2 27 253 125 1114 729 3515 2187 1026

Table 1: Number of function evaluation required to achieve a certain degreeof polynomial exactness using tensor product and sparse grid formulations.The parameter l is called the level and is directly connected to the accuracyof the interpolant on tensor grids; on sparse grids the relationship is morecomplex.

construct quadrature rules in the three-dimensional parameter space (ξ1, ξ2, ξ3). InTable 1 the number of function evaluations (abscissas) required to achieve a certainorder of polynomial exactness are reported.

The expectation and variance of the wall temperature are reported in Fig. 5.2and 5.2, respectively.

In this application, it is clear that accurate temperature statistics can be ob-tained efficiently using stochastic collocation with respect to Monte Carlo sampling.The difference between tensor product and sparse grid formulations is very small.

5.3 Shock dominated problems

Uncertainty propagation algorithms based on polynomial basis are particularly ef-fective when the output of interest are smoothly varying with respect to the inputuncertainties. The presence of discontinuities in the response surface poses a chal-lenge; in physical terms, these discontinuities represent sharp system transitionssuch as the occurrence of flow separation or shock interactions.

Consider the problem governed by the non-homogenous Burgers equations [27]:

∂u

∂t+

∂

∂x(u2

2) =

∂

∂x(sin2 x

2), 0 ≤ x ≤ π, t > 0

with the initial condition u(x, 0) = β sin x, and boundary conditions u(0, t) =u(π, t) = 0. β represents the uncertainty in the initial condition and is definedas

β =−1 +

√

1 + 4σ2ξ

2σ2ξ

where ξ is a Gaussian random variable and σ is used to control the amount ofuncertainty.

The exact steady solution contains a shock located at xs = f(β) and the desiredoutput is the PDF of the shock location. The response surface u as a function of βis reported in Fig. 12.


17 - 22 RTO-EN-AVT-162

−0.5 0 0.5

80

100

120

140

160

180

x1

E[T

wal

l(x1)]

M:10,0002:273:1254:729

−0.5 0 0.5

80

100

120

140

160

180

x1

E[T

wal

l(x1)]

M:10,0002:253:1114:351

Figure 9: Expectation of the wall temperature. Tensor product (left) ver-sus sparse grids (right). Monte Carlo (MC) statistics are based on 10,000samples; multiple quadrature formulas are reported for different levels (seeTable 1).

−0.5 0 0.5

50

100

150

200

250

300

x1

Var

[Tw

all(x

1)]

M:10,0002:273:1254:729

−0.5 0 0.50

50

100

150

200

250

300

x1

Var

[Tw

all(x

1)]

M:10,0002:253:1114:351

Figure 10: Variance of the wall temperature. Tensor product (left) versussparse grids (right). Monte Carlo (MC) statistics are based on 10,000 sam-ples; multiple quadrature formulas are reported for different levels (see Table1).


RTO-EN-AVT-162 17 - 23

100 120 140 160 180 200 220 240

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Tstag

pdf T

stag

M:10,000

2:27

3:125

4:729

100 120 140 160 180 200 220 240 260

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Tstag

pdf T

stag

M:10,000

2:25

3:111

4:351

Figure 11: Probability distribution function of the wall temperature at thestagnation point. Tensor product (left) versus sparse grids (right). MonteCarlo (MC) statistics are based on 10,000 samples; multiple quadratureformulas are reported for different levels (see Table 1).

Figure 12: Solution of the in-homogeneous Burgers equations with uncertaininitial condition. The sharp transition corresponds to a shock whose locationis a function of the input uncertainty.


17 - 24 RTO-EN-AVT-162

Figure 13: Probability distribution of the shock location for the inhomo-geneous Burgers equations with uncertain initial condition. Low , σ = 0.1(left) and high input variability, σ = 0.6 (right). Results obtained usinghigh-order stochastic Galerkin expansions [28]

Hestevan et al. [25] have applied the stochastic Galerkin approach to studythis problem; high order expansions in Hermite polynomials were used to computethe PDF of the shock location. The availability of exact PDFs allows to identifythe limitation of the numerical approach. In Fig. 5.3 the solution is reported forlow and high degree of input variability (σ set to 0.1 and 0.6 respectively). Forthe low variability case the computed results are in reasonable agreement withthe exact PDF, but some oscillations are present. For the high variability case,the oscillations are the dominant feature and the shape of the PDF is completelymisrepresented. In this case, the problem is that the solution is discontinuous andthe polynomial basis (underlying the stochastic expansion) is highly oscillatory -Gibbs phenomenon.

An alternative response reconstruction has been developed to handle strongnon-linear or discontinuous surfaces. The method is based on a Pade-Legendreinterpolatory formula [32]; it is a stochastic collocation approach that eliminatesthe Gibbs phenomenon. Assume N solution evaluations are available; given theintegers M and L, the pair of Legendre polynomials P and Q of order less orequal than M and L, respectively, are said to be the solution of the (N, M, L)Pade-Legendre interpolation problem [26] of u if

〈P − Qu, φ〉 = 0 (12)

and the rational function R(u) := P/Q is defined as the approximation of u. Thedetails of the construction of R are reported in Chantrasmi et al. [32]; here it issufficient to observe that if u is discontinuous, the polynomial Q can be regarded


RTO-EN-AVT-162 17 - 25

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

shock location

PD

F

Exact PDF

Pade−Legendre

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

shock location

PD

F

Exact PDF

Pade−Legendre

Figure 14: Probability distribution of the shock location for the inhomo-geneous Burgers equations with uncertain initial condition. Low , σ = 0.1(left) and high input variability, σ = 0.6 (right). Results obtained using thePade-Legendre stochastic collocation approach [32]

as a preconditioner (or a filter) such that u ·Q is smooth and can be efficiently andaccurately interpolated by the polynomial P .

Results obtained using the Pade-Legendre stochastic collocation approach arereported in Fig. 5.3 in terms of the PDF of the shock locations. In this case theaccuracy is preserved even for high input variability.

6 Conclusions and outlook

The growing interest in uncertainty quantification has motivated the developmentof a variety of mathematical approaches. In particular, probabilistic uncertaintypropagation methods have received considerable attention.

The choice of the appropriate method to use for a specific application is notobvious. For typical fluid mechanics simulations some common considerations are:

• expensive function evaluation: sampling based methods are typically not ap-propriate because they might require several thousand full computations tobuild the statistics of the outputs;

• large number of uncertainties: boundary conditions, material properties, ge-ometry specification, etc. introduce many independent input parameters thathave to be characterize. Methods that suffer from curse of dimensionalityquickly become unfeasible;

• non-linear system responses: transitions and bifurcations are typical of fluidmechanics, especially for compressible flows. Methods that strictly require asmooth dependency between inputs and outputs can be ineffective.


17 - 26 RTO-EN-AVT-162

In spite of these difficulties, probabilistic UQ is naturally amenable to mathe-matical and numerical analysis and, therefore, is expected to achieve a high levelof maturity in the near future.

In addition to the methods presented here several other methods have beenapplied especially in the field of structural mechanics. It is also worth mentioningthat alternative approaches not based on probabilistic reasoning have been proposedand used with some success. It is not generally clear when probabilistic methodsfail or are insufficient; the treatment of epistemic uncertainty remains difficult andpossibly the greatest challenge in uncertainty quantification.

Acknowledgments

This work is supported by the Department of Energy within the Advanced Simula-tion and Computing Program. The notes are based on work carried out at Stanfordby Paul Constantine and Tonkid Chantrasmi.

References

[1] V. Barthelmann, E. Novak, K. Ritter, High dimensional polynomial interpo-lation on sparse grids, Adv. Comput. Math. 12 (2000) 273-288.

[2] Bowman, A. W., A. Azzalini, Applied Smoothing Techniques for Data Analy-sis, Oxford University Press, 1997.

[3] B. Ganapathysubramanian, N. Zabaras, Sparse grid collocation schemesfor stochastic natural convection problems, J. Comput. Phys. (2007),doi:10.1017/j.jcp.2006.12.014

[4] W. Gautschi, Algorithm 726: ORTHOPOL-A package of routines for gener-ating orthogonal polynomials and Gauss-type quadrature rules, ACM Trans.Math. Software20, (1994) 2162.

[5] R. Ghanem and A. Doostan, On the construction and analysis of stochasticmodels: characterization and propagation of the errors associated with limiteddata, J. of Comput. Phys. 217 (2006) 6381.

[6] E. Isaacson, H.B. Keller, Analysis of Numerical Methods, Dover, 1994.

[7] A. S. Kronrod, Nodes and Weights of Quadrature Formulas. Consultants Bu-reau, New York, 1965

[8] M. Loeve, Probability Theory, 4th ed., Springer-Verlag, New York, 1997.

[9] F. Nobile, R. Tempone and C.G. Webster. A sparse grid stochastic colloca-tion method for partial differential equations with random input data. Reportnumber: TRITA-NA 2007:07. To appear SINUM.

[10] E. Novak, K. Ritter, Simple cubature formulas with high polynomial exactness,Constructive Approx., 15 (1999) 499-522.


RTO-EN-AVT-162 17 - 27

[11] E. Novak, K. Ritter, High dimensional integration of smooth functions overcubes, Numer. Math. 75 (1996) 79-97.

[12] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. NumericalRecipes in C. Cambridge University Press, 2nd ed., 1992.

[13] J.A. Rice, Mathematical Statistics and Data Analysis, 2nd ed., Duxbury Press,1995.

[14] B.D. Ripley, Stochastic Simulation, 1st ed., Wiley, 1987.

[15] S.A. Smolyak, Quadrature and interpolation formulas for tensor products ofcertain classes of functions, Soviet math. Dokl., 4 (1963) 240-243.

[16] D.M. Tartakovsky, D. Xiu, Stochastic analysis of transport in tubes with roughwalls, J. Comput. Phys. 217 (2006) 248-259.

[17] L.N. Trefethen, Is Gauss quandrature better than Clenshaw-Curtis? SIAMReview, to appear, 2007.

[18] J.H. VanLint, R.M. Wilson, A Course in Combinatorics, 2nd ed., CambridgeUniversity Press, 2001.

[19] D. Xiu, J.S. Hesthaven, High order collocation methods for the differentialequations with random inputs, SIAM J. Sci. Comput. 27 (2005) 1118-1139.

[20] G.W. Wasilkowski and H. Wozniakowski, Explicit cost bounds of algorithmsfor multivariate tensor product problems, J. of Complexity, 11:1-56 1995.

[21] D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for

stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619–644.

[22] R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach,Springer-Verlag, New York, 1991.

[23] N. Wiener, The homogeneous chaos, Amer. J. Math., 60 (1938), pp. 897–936.

[24] Szego Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc.,1975.

[25] J. S. Hesthaven , S. M. Kaber , L. Lurati, Pad-Legendre Interpolants forGibbs Reconstruction, Journal of Scientific Computing, v.28 n.2-3, pp. 337-359, September 2006

[26] G. A. Baker, Pade Approximants, Cambridge Univ Pr.

[27] M.D. Salas, S. Abarbanel and D. Gottlieb, Multiple steady states for charac-teristic initial value problems, Applied Numerical Mathematics 2 (1986), pp.193-210.

[28] Q.-Y. Chen, D. Gottlieb, J. S. Hesthaven, Uncertainty analysis for the steady-state flows in a dual throat nozzle, Journal of Computational Physics, v.204(1), pp. 378-398, March 2005


17 - 28 RTO-EN-AVT-162

[29] J.C. Helton and F.J. Davis, Sampling-based Methods, in Sensitivity Analysis,A. Saltelli, C. Chan, and E.M. Scott, Eds. New York: Wiley, 2000, pp. 101-153.

[30] J.C. Helton and F.J. Davis, Latin hypercube sampling and the propagationof uncertainty in analyses of complex systems, Reliability Engineering andSystem Safety, vol. 81, pp. 23-69, 2003.

[31] Z. Sandor and P. Andras, ”Alternative Sampling Methods for Estimating Mul-tivariate Normal Probabilities”, Econometric Institute Report EI 2003,05

[32] T. Chantrasmi, A. Doostan and G. Iaccarino, Pade-Legendre Approximantsfor Uncertainty Analysis with Discontinuous Response Surfaces, submitted toJCP, 2008

[33] W.L. Oberkampf and M. F. Barone, Measures of agreement between compu-tation and experiment: validation metrics. Journal of Computational Physics,V. 217, 2006, pp. 5–36.

[34] Knupp, P., and Salari, K. (2003), Verification of Computer Codes in Com-putational Science and Engineering, Chapman and Hall/CRC, Boca Raton,FL.

[35] AIAA (1998), ”Guide for the Verification and Validation of ComputationalFluid Dynamics Simulations,” American Institute of Aeronautics and Astro-nautics, AIAA-G-077-1998, Reston, VA.

[36] R. W. Walters and L. Huyse, Uncertainty Analysis for Fluid Mechanics withApplications, NASA/CR 2002-211449.

[37] S. Fersin, R. B. Nelsen, et al. Dependence in probabilistic modeling, Dempster-Shafer theory, and probability bounds analysis, SAND Report 2004-3072.

[38] Oberkampf, W.L. and J. C. Helton. (2002). Investigation of evidence theory forengineering applications. AIAA Non-Deterministic Approaches Forum, April2002, Denver, Colorado, paper 2002-1569.


RTO-EN-AVT-162 17 - 29


17 - 30 RTO-EN-AVT-162

rto-en-avt-162 17 - 1 · 13. supplementary notes see also ada562449. rto-en-avt-162,...

Documents